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[attr] lfs-file filter=lfs diff=lfs merge=lfs -text
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*.pdf lfs-file
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+13
@@ -0,0 +1,13 @@
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# Python-generated files
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__pycache__/
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*.py[oc]
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build/
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dist/
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wheels/
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*.egg-info
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# Virtual environments
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.venv
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#
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.vscode
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@@ -0,0 +1 @@
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3.12
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@@ -0,0 +1,273 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "8f8630bf",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9751d328",
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"metadata": {},
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"source": [
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"# Floating Point\n",
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"\n",
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"**Double Precision Floating Point Number Representation**\n",
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"\n",
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"$$(-1)^s \\times 2^{c-1023} \\times (1+f)$$\n",
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"\n",
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"* $s$: sign bit\n",
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"* $c$: 11 bit exponent\n",
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"* $f$: 52 bit fraction"
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]
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},
|
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{
|
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"cell_type": "markdown",
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"id": "9c0b5413",
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"metadata": {},
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"source": []
|
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},
|
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{
|
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"cell_type": "markdown",
|
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"id": "964d247f",
|
||||
"metadata": {},
|
||||
"source": []
|
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},
|
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{
|
||||
"cell_type": "markdown",
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"id": "6f7aef85",
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"metadata": {},
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"source": [
|
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"# Root Finding\n",
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"\n",
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"When there is no or hard to find analytic solution.\n",
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"\n",
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"* General procedure of root finding\n",
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"\n",
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"\n",
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"## Incremental Search\n",
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"\n",
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"Detecting a sign change in the interval $[x, dx]$\n",
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"\n",
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"\n",
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"## Bracketing Method\n",
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"\n",
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"* there must be a solution in a given interval $[a,b]$\n",
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"* reducing the interval systematically\n",
|
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"* [Merit]Convergence to an exact solution\n",
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"\n",
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"Methods:\n",
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"* bisection\n",
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"* linear interpolation\n",
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"\n",
|
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"## Open Method\n",
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"\n",
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"* starting from an arbitrary point, find the root based on a regular iteration\n",
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"* [Risk] divergence\n",
|
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"* [Merit] faster than _bracketing_\n",
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"\n",
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"Methods:\n",
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"* fixed-point iter\n",
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"* Newton-Raphson method\n",
|
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"* secant method\n",
|
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"* Muller method\n"
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]
|
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},
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{
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"cell_type": "markdown",
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"id": "2649c3d4",
|
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"metadata": {},
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"source": [
|
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"## Bisection Method\n",
|
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"based on the principle: **A solution must exists in $[x_1, x_2]$ if $f(x_1)f(x_2) < 0$**\n",
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"\n",
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"so interval reduces by half at each iteration.\n",
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"\n",
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"* simple\n",
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"* maximum error $<$ (initial interval/$2^n$)\n",
|
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"* slow converge\n",
|
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"* cannot cope with multiple roots"
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]
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},
|
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{
|
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"cell_type": "markdown",
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"id": "ef80f29d",
|
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"metadata": {},
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"source": [
|
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"## Linear Interpolation Method(False Position Method)\n",
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"\n",
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"An interval $[a_{n+1}, b_{n+1}]$, containing an approximation of $f(x) = 0$, is found from an interval $[a_n, b_n]$\n",
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"\n",
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"$$p_{n+1} = a_n - \\frac{f(a_n)(b_n-a_n)}{f(b_n) - f(a_n)}$$\n",
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"\n",
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"then set $$\\begin{cases}\n",
|
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"a_{n+1} = a_n, b_{n+1} = p_{n+1} &\\quad \\text{if} f(a_n)f(p_{n+1}) < 0\\\\\n",
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"a_{n+1} = p_{n+1}, b_{n+1} = b_n &\\quad \\text{else}\n",
|
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"\\end{cases}$$\n",
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"\n",
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"* faster convergence than bisection\n",
|
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"* convergence speed depends on **curvature**\n",
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"\n",
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"\n",
|
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"\n",
|
||||
"### Modified Linear Interpolation\n",
|
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"\n"
|
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]
|
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},
|
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{
|
||||
"cell_type": "markdown",
|
||||
"id": "856f5bbf",
|
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"metadata": {},
|
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"source": [
|
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"## Newton-Raphson Method\n",
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"\n",
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"\n",
|
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"* **fast convergence**: quadratic convergence\n",
|
||||
"* Inefficient if **the derivative calculation is complex**\n",
|
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"* Initial guess matter\n",
|
||||
" * Cycling\n",
|
||||
" * Wandering\n",
|
||||
" * Overshooting(a root the methods get is not the root I wanted)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
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"id": "cf78e848",
|
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"metadata": {},
|
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"source": [
|
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"## Secant Method\n",
|
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"\n",
|
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"$$p_{n+1} = p_n - \\frac{f(p_n)(p_n - p_{n-1})}{f(p_n) - f(p_{n-1})}$$\n",
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"\n",
|
||||
"Similar to linear interpolation but update rule is different.\n",
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"\n",
|
||||
"* fast convergence than **linear interpolation**\n",
|
||||
"* more stable than Newton-Raphson\n",
|
||||
" * but convergence is slower than Newton-Raphson"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 36,
|
||||
"id": "167cae2f",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 640x480 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"x = np.linspace(0, 4)\n",
|
||||
"\n",
|
||||
"f = lambda t: t**2 - 5\n",
|
||||
"y = f(x)\n",
|
||||
"\n",
|
||||
"p = [0.5, 3.5]\n",
|
||||
"\n",
|
||||
"for _ in range(2):\n",
|
||||
" p.append(p[-1] - f(p[-1]) * (p[-1] - p[-2]) / (f(p[-1]) - f(p[-2])))\n",
|
||||
"\n",
|
||||
"p = np.array(p)\n",
|
||||
"plt.plot(\n",
|
||||
" p,\n",
|
||||
" np.zeros_like(p),\n",
|
||||
" \"ro\",\n",
|
||||
")\n",
|
||||
"for i, p0 in enumerate(p):\n",
|
||||
" plt.text(p0 + 0.05, 0.1, f\"$p_{i}$\", size=16, color=\"black\", va=\"bottom\")\n",
|
||||
"\n",
|
||||
"plt.plot(p, f(p), \"ro\")\n",
|
||||
"\n",
|
||||
"for pos in p:\n",
|
||||
" _y = [0, f(pos)]\n",
|
||||
" plt.vlines(pos, min(_y), max(_y), color=\"red\")\n",
|
||||
"\n",
|
||||
"for p0, p1 in zip(p[:-1], p[1:]):\n",
|
||||
" plt.plot([p0, p1], [f(p0), f(p1)], color=\"green\")\n",
|
||||
"\n",
|
||||
"plt.title(\"Secant Method\")\n",
|
||||
"plt.plot(x, y, color=\"blue\")\n",
|
||||
"plt.hlines(0, 0, 4, color=\"black\")\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "17e94fbf",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Fixed Point Iteration\n",
|
||||
"\n",
|
||||
"principle: $f(x) = 0 \\to x = g(x)$\n",
|
||||
"\n",
|
||||
"$$x_{k+1} = g(x_k)$$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "4cc731ad",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Muller Method"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c3702bdd",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Error Analysis"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c0e38d45",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Accelerating Convergence\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1e7717a7",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Fail-safe Methods\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.12.11"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
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@@ -0,0 +1,11 @@
|
||||
[project]
|
||||
name = "2025-02-numerical"
|
||||
version = "0.1.0"
|
||||
description = "Add your description here"
|
||||
readme = "README.md"
|
||||
requires-python = ">=3.12"
|
||||
dependencies = [
|
||||
"jupyter>=1.1.1",
|
||||
"matplotlib>=3.10.6",
|
||||
"numpy>=2.3.2",
|
||||
]
|
||||
Reference in new issue
Block a user